Asymmetric Orbifold toward Heterotic Anomalous U(1) GUT Toshifumi Yamashita　 (Nagoya University) 09 Nov. 2009　@KEK mini workshop working with 　　 Nagoya U. : Maekawa, Moriyama, Teraguchi Takei, Ito, Kuwakino 　　 NCTU (Taiwan) :　Keijiro Takahashi

Introduction anomalous U(1) GUT GUTs from the Heterotic string. N. Maekawa & T.Y. & ・・・ (’01-’04) an interesting SUSY-GUT e.g. E6*SU(2)H DTS, PD, Yukawa hierarchy, SUSY flavor etc. anomaly : assumed to be cancelled by GS. to be checked We aim to embed it into string. Asymmetric orbifold GUTs from the Heterotic string. examined in '80 exhaustively simple cpt. no adjoint Higgs Kac-Moody level

Plan Introduction anomalous U(1) GUT overview Narain cpt. & Modular inv. & orbifold Models Summary

Grand Unified Theories Unification of Gauge Group : non-Abelian Charge Quantization Unification of Matter SU(5) SO(10) E6 + + = + + = wrong GUT relation

Grand Unified Theories sector in E6 model E-twisting M.Bando and T.Kugo mass massless : all come from and . mild hierarchy in sector !! large mixing

Grand Unified Theories Unification of Gauge Group : non-Abelian Charge Quantization How about Higgs ? Proton decay Doublet-Triplet (DT) Splitting Problem

SUSY-GUT fascinating extension of SM hierarchal Yukawa? unifications of forces and of matter charge quantization stabilization of the weak scale gauge coupling unification (GCU) Higgs? SUSY, (FCNC)? Proton decay?

Anomalous U(1) gauge symmetry Low energy theory of string theory Anomaly Green-Schwartz mechanism Froggatt-Nielsen (FN) mechanism : FN field : cutoff scale ex)

Anomalous U(1) gauge symmetry Low energy theory of string theory Anomaly Green-Schwartz mechanism Froggatt-Nielsen (FN) mechanism SUSY-zero (holomorphic-zero) mechanism superpotential : holomorphic does not appear in . Only positively charged operators can appear. This strongly constrain !!

Anomalous U(1) gauge symmetry Low energy theory of string theory Anomaly Green-Schwartz mechanism Froggatt-Nielsen (FN) mechanism SUSY-zero (holomorphic-zero) mechanism Ex) Adjoints of SO(10) : Only the linear terms in A’ are relevant for EOM.

Anomalous U(1) gauge symmetry Low energy theory of string theory Anomaly Green-Schwartz mechanism Froggatt-Nielsen (FN) mechanism SUSY-zero (holomorphic-zero) mechanism Ex) Adjoints of SO(10) : The # of non-zero xi labels the vacuum N=3 : Dimopoulos-Wilczek VEV DTS

Anomalous U(1) GUT SUSY-GUT w/ anomalous U(1) symmetry Ansatz N.Maekawa & T.Y. & … SUSY-GUT w/ anomalous U(1) symmetry Ansatz “generic” interaction introduce all possible interactions (including NROs) assume all the coefficients are O(1) vacuum (cutoff scale) : GUT singlet operator, : U(1)A charge : breaking scale of U(1)A charge assignment ＝ def. of model

Anomalous U(1) GUT Generic interaction GCU FN mechanism N.Maekawa & T.Y. & … Generic interaction FN mechanism SUSY-zero mechanism Natural Gauge Coupling Unification hierarchal Fermion Yukawa DW type of adjoint VEV DT Splitting proton decay GCU

Anomalous U(1) GUT fascinating extension of SM N.Maekawa & T.Y. & … hierarchal Yukawa? fascinating extension of SM unifications of forces and of matter charge quantization stabilization of the weak scale gauge coupling unification (GCU) Higgs? SUSY, (FCNC)? horizontal symm. Proton decay? Almost all the problem can be solved!!

Anomalous U(1) GUT examples of E6 model two charged adjoint N.Maekawa & T.Y. & … examples of E6 model PTP 107, 1201 (2002) N.Maekawa and T.Y. E6 * Z2 PLB561, 273(2003) N.Maekawa two charged adjoint 6 – 3 or 5 – 2 generations E6 * SU(2)

Plan Introduction anomalous U(1) GUT overview Narain cpt. & Modular inv. & orbifold Models Summary

related to doubled geometry? overview anomalous U(1) GUT -- E6 model charged adjoint Higgs higher Kac-Moody level Het, M, F, … related to doubled geometry? diagonal embedding asymmetric orbifold -- Narain cpt. asymmetric treatment of left- and right-moving string need careful consistency check

overview Kac-Moody algebra Diagonal embedding : structure constant ex) mode exp. of world sheet current Diagonal embedding w/

overview # of generation model cpt. of or SO(32) Hetero w/ Wilson line e.g. J.Erler (1996) (in known models) Z.Kakushadze & H.Tye (1996) model cpt. of or SO(32) Hetero w/ Wilson line Z.Kakushadze & H.Tye (1996) claimed w/ adjoint Higgs, 3 generations & non-Abelian Hidden gauge Only one model is possible. Narain compactification (flat instead of CY, for simplicity)

Plan Introduction anomalous U(1) GUT overview Narain cpt. & Modular inv. & orbifold Models Summary

Narain cpt. & Modular inv. & orbifold string w/ compactification

Narain cpt. & Modular inv. & orbifold Heterotic string from modular inv. left right 4D 6D 16D "compactified" on or Spin(32)/Z2 Lie lattice here? no geometrical interpretation Narain compactification discard geometric interpretation also in cpt. 6D gauge symm w/ rank 22 is possible : ex)

Narain cpt. & Modular inv. & orbifold Modular invariance red blue shift The modular transformations, closed string 1-loop amp. do not change the torus. The amplitude should be invariant under these tr..

Narain cpt. & Modular inv. & orbifold partition function : NS : R

Narain cpt. & Modular inv. & orbifold partition function even T S self-dual (or )

Narain cpt. & Modular inv. & orbifold geometric compactification ex.) A2 lattice winding momentum This momentum lattice is actually even & self-dual. proof

Narain cpt. & Modular inv. & orbifold Narain lattice Cf.) geometric : simple roots of a (simply laced) Lie group. winding momentum R=1 21/2 ( ; 0) R= 2-1/2 for pR=0 : Cartan matrix This lattice leads to gauge symmetry. This does not, except for SU(2).

Narain cpt. & Modular inv. & orbifold ESDL generating technique left right

Narain cpt. & Modular inv. & orbifold ESDL generating technique more left right left right

Narain cpt. & Modular inv. & orbifold ZN for simplicity diagonal embedding 4D N =1 SUSY Manifold / discrete symm. impose identity under (discrete) orbifold action twist (rotation) shift (reflection) ex.) A2 lattice : Z2,3,6 1/3 twist : does not change the lattice

Narain cpt. & Modular inv. & orbifold ZN for simplicity diagonal embedding 4D N =1 SUSY Manifold / discrete symm. impose identity under (discrete) orbifold action twist (rotation) shift (reflection) ex.) A2 lattice : Z2,3,6 additional string states : does not change the lattice

Narain cpt. & Modular inv. & orbifold partition function additional string state w/ different B.C. S Modular tr. S S2=1 T

Narain cpt. & Modular inv. & orbifold partition function ex) Z3 T : S : S Modular tr. S S2=1 T

Narain cpt. & Modular inv. & orbifold partition function ex) Z3 T : S : Rules on are established for symm. orbifold. Modular inv. P.F. is inv. if each sector satisfies the tr. property. For asymm. orbifold, not yet. Z.Kakushadze & H.Tye (1996)

Narain cpt. & Modular inv. & orbifold partition function ex) Z3 T : S : Modular inv. We make and then define by S, T.

Narain cpt. & Modular inv. & orbifold partition function ex) Z3 T : S : Modular inv. The remaining S, T give non-trivial conditions.

Plan Introduction anomalous U(1) GUT overview Narain cpt. & Modular inv. & orbifold Models Summary

Models 4D SUSY 4D left right 4D N =1 SUSY zero modes SO(8)LC repr. phase even

combined with phaseless RM to form gauge field. Models diagonal embedding : Z3 4D left E6 E6 E6 combined with phaseless RM to form gauge field. right level 3 E6

Models lattice for right-mover E6 A2 A2 A2 D4 A2 4D left right 4D N =1 SUSY : Z3 E6 A2 A2 A2 D4 A2

Models possible models A2 A2 E6 E6 E6 E6 A2 A2 E6 E6 E6 A2 A2 A2 D4 Z.Kakushadze & H.Tye (1996) left A2 A2 E6 E6 E6 right E6 1 neutral adj. 5 – 2 generations Z6 A2 A2 E6 E6 E6 A2 A2 A2 Z6 : vector-like Z3 : 9 generations D4 E6 E6 E6 D4 A2

Summary We are working on Heterotic GUTs to derive Anomalous U(1) GUT from string. to get 4D N =1 SUSY E6 model w/ adjoint Higgs & 3 generations E63 lattice via Narain compactification diagonal embedding in asymmetric orbifold Future works We want two charged Adjoint Higgs towards Anomalous U(1) GUT

Back-up Slides

Narain cpt. & Modular inv. & orbifold ESDL generating technique (248) (78,1)+(1,8)+[(27,3)+c.c.] (78;1)+(1;78)+[(27;27)+c.c.] left (78;1,1,1)+(1;8,1,1)+[(1;3,3,3)+c.c.] +[(27;3,3,1)+c.c.] right (78,1,1;1)+(1,1,1;8)+[(1,27,27;3)+c.c.] +[(27,27,27;1)+(27,1,27;3)+(27,27,1;3) +(27,27,27;1)+(27,1,27;3)+(27,27,1;3)] left right

Narain cpt. & Modular inv. & orbifold ESDL generating technique (248) (78,1)+(1,8)+[(27,3)+c.c.] (78;1)+(1;78)+[(27;27)+c.c.] left (78;1,1,1)+(1;8,1,1)+[(1;3,3,3)+c.c.] +[(27;3,3,1)+c.c.] right (78,1,1;1)+(1,1,1;8)+[(1,27,27;3)+c.c.] +[(27,27,27;1)+(27,1,27;3)+(27,27,1;3) +(27,27,27;1)+(27,1,27;3)+(27,27,1;3)] left right